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The idea is start at the left and work towards the right. Thus if the answer is α = a−2 /32 + a−1 /3 + a0 + a1 3 + · · · , then a−2 = 2, a−1 = 1, a0 = 2 + 1 = 0 + 1 · 3 ≡ 0, 3 and so a1 = 2 + 2 + 1 = 2 + 1 · 3 ≡ 2 3 where the 1 is carried from the 30 term. Continuing we get a2 = 0 + 1 + 1 = 2, a3 = 2 + 1 = 0 + 1 · 3 ≡ 0, 3 and so we get α = 2/32 + 1/3 + 0 + 2 · 3 + 2 · 32 + 0 · 33 + · · · as the sum to within a term of 3-adic norm smaller than 1/33 . Notice that the p-adic expansion of a p-adic number is unique, whereas the decimal expansion of a real need not be.

For the image of ω, we remark that the distinct numbers in the list 0, 1, 2, . . , p − 1 satisfy |r − s|p = 1. If r ̸= s, then |ω(r) − ω(s)|p = 1. Hence, the image of the function ω has at least p distinct elements, all of which are roots in Qp of X p − X. As Qp is a field, there are not more than p of these roots. So this polynomial factors as X p − X = X(X − ω(1))(X − ω(2)) · · · (X − ω(p − 1)) and the p roots are the only elements in the image of ω. 14. For the prime p = 2, the roots of X 2 − X are 0,1.

As in real analysis, we can investigate for which values of x this converges or diverges. 7. Take αn = 1 for all n. Then  = 0 lim(p) xn n→∞  1 if |x|p < 1, otherwise. ∑ n So this series converges if and only if |x|p < 1. Of course, in R the series x converges if |x| < 1, diverges if |x| > 1, diverges to +∞ if x = 1 and oscillates through the values 0 and −1 if x = −1. 8. For the series ∑ nxn , we have |x|p n |nxn |p = |n|p |xn |p which tends to 0 in R if |x|p < 1. So this series certainly converges for every such x.

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